The perimeter of the ellipse is at least the same as the perimeter of the matching armscye on the body of the garment. If the perimeter of the ellipse is smaller than the armscye, it won’t “fit” – used in cap sleeves only. If the perimeter of the ellipse is larger than the armscye, you can ease the excess all around, or into the top to produce a gather. In any event, the armscye perimeter can be calculated from the garment body measurements.

You also will have a measurement for top arm circumference. The top arm circumference for a woman of average height is around

Bust

32

36

40

44

48

Top Arm Circumf.

10.5

11.5

13

14

15.5

Don’t forget to add ease! This is an unclothed measurement.

You can see from the diagram below, that the top arm circumference, i.e. the widest part of the sleeve, is equivalent to the width of two ellipses (0.5 + 1 + 0.5 ellipse widths). Hence, the horizontal radius of the sleeve cap ellipse is a quarter of the top arm circumference.

The sleeve cap height is twice the vertical radius of the ellipse, i.e. the same as the height of one ellipse.

If you plug the values for perimeter and horizontal radius into Ramanujan’s Approximation for the perimeter of an ellipse, you can get a good-enough value for the sleeve cap height.

The sleeve cap height is twice the value for c, the vertical radius of the ellipse, approximated by

Where p is the armscye perimeter / pi

And w is the top arm circumference / 4

The approximation above is essentially Ramanujan’s Approximation expressed in terms of one of the radii.